We study Novikov superalgebras with nondegenerate associative supersymmetric bilinear forms which are called symmetric Novikov superalgebras. We show that Novikov symmetric superalgebras are associative superalgebras with additional condition. Several examples of symmetric Novikov superalgebras are included, in particular, examples of symmetric Novikov superalgebras which are not 2-nilpotent. Finally, we introduce some notions of double extensions in order to give inductive descriptions of symmetric Novikov superalgebras.

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e. even or odd) and associative bilinear form on A. In this paper, we show that any associative superalgebra with non null product can not admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descriptions of even-symmetric associative superalgebras and odd-symmetric associative superalgebras. We obtain also an other interesting description of odd-symmetric associative superalgebras whose even parts are semi-simple bimodules without using the notions of double extensions.

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The purpose of this paper is to study quadratic color Hom-Lie algebras. We present some constructions of quadratic color Hom-Lie algebras which we use to provide several examples. We describe T * -extensions and central extensions of color Hom-Lie algebras and establish some cohomological characterizations. IntroductionThe aim of this paper is to introduce and study quadratic color Hom-Lie algebras which are graded Hom-Lie algebras with ε-symmetric, invariant and nondegenerate bilinear forms. Color Lie algebras, originally introduced in [26] and [27], can be seen as a direct generalization of Lie algebras bordering Lie superalgebras. The grading is determined by an abelian group Γ and the definition involves a bicharacter function. Hom-Lie algebras are a generalization of Lie algebras, where the classical Jacobi identity is twisted by a linear map. Quadratic Hom-Lie algebras were studied in [9]. Γ-graded Lie algebras with quadratic-algebraic structures, that is Γ-graded Lie algebras provided with homogeneous, symmetric, invariant and nondegenerate bilinear forms, have been extensively studied specially in the case where Γ = Z 2 (see for example [2,6,7,8,11,24,28]). These algebras are called homogeneous (even or odd) quadratic Lie superalgebras. One of the fundamental results connected to homogeneous quadratic Lie superalgebras is to give its inductive descriptions. The main tool used to obtain these inductive descriptions is to develop some concept of double extensions. This concept was introduced by Medina and Revoy (see [24]) to give a classification of quadratic Lie algebras. The concept of T * -extension was introduced by Bordemann [? ]. Recently a generalization to the case of quadratic (even quadratic) color Lie algebras was obtained in [25] and [33]. They mainly generalized the double extension notion and its inductive descriptions.Hom-algebraic structures appeared first as a generalization of Lie algebras in [1,12,13] were the authors studied q-deformations of Witt and Virasoro algebras. A general study and construction of Hom-Lie algebras were considered in [18], [20]. Since then, other interesting Hom-type algebraic structures of many classical structures were studied as Hom-associative algebras, Hom-Lie admissible algebras and more general G-Hom-associative algebras [21], n-ary Hom-Nambu-Lie algebras [4], Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras [22], Hom-alternative algebras, Hom-Malcev algebras and Hom-Jordan algebras [15,22,36]. Hom-algebraic structures were extended to the case of Γ-graded Lie algebras by studying Hom-Lie superalgebras and Hom-Lie admissible superalgebras in [5]. Recently, the study of Hom-Lie algebras provided with quadratic-algebraic structures was initiated by S. Benayadi and A. Makhlouf in [9] and our purpose in this paper is to generalize this study to the case of color Hom-Lie algebras.

We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [2]. We use this concept to give an inductive description of nilpotent homogeneoussymplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than evenquadratic even-symplectic considered in [6]. We study the structures of even (resp. odd)-quadratic odd (resp. even)-symplectic Lie superalgebras and oddquadratic odd-symplectic Lie superalgebras and we give its inductive descriptions in terms of quadratic generalized double extensions and odd quadratic generalized double extensions. This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [6]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superargebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [6].